It appears to be a relic of Middle English usage:
The following answer from forum.wordreference.com appears to shed some light on the usage of “me instead of my:”
In Shakespeare's time (around 1600) and prior to that, my (and thy) would have a strong emphatic form with the vowel /aɪ/ and a weak form /ɪ/. This is similar to how we have the strong form of he /hiː/ together with the weak forms /hɪ/ or /ɪ/.
Since that time, Standard English has moved towards using the strong form /maɪ/ in all contexts, whereas many regional varieties in the UK still have the weak form /mɪ/. When writers want to indicate this pronunciation in writing, they often resort to spelling this me, whereas in reality the speakers are actually saying my.
And actually, as Wiktionary shows, mi, meaning my, was used in Middle English:
my, mie, me
Apocopated form of min, myn, from Old English mīn (“my, mine”), from Proto-Germanic *mīnaz (“my, mine”, pron.) (genitive of *ek (“I”)), from Proto-Indo-European *méynos (“my; mine”).
(unstressed) IPA(key): /mi/
mi (nominative I)
1- First-person singular genitive determiner: my
mi is usually used before a consonant (other than h-), while min is usually used before a vowel or h-, much as with Modern English an vs a.